Symmetry vector fields on the 4-dimensional Riemannian and Lorentzian Lie group Sol^4_0

Abstract

In this paper, we investigate various symmetry vector fields on the 4-dimensional Riemannian and Lorentzian Lie group \( Sol_0^4 \). Using Lie derivatives, we characterize affine vector fields with respect to the Levi-Civita connection. We further obtain Ricci collineations vector fields, curvature collineations vector fields, and matter collineations vector fields by computing the Lie derivatives of the Ricci tensor, Riemann curvature tensor, and energy-momentum tensor, respectively.  These results contribute to the understanding of geometric and physical symmetries on \( Sol_0^4 \), relevant in both mathematical physics and differential geometry.